Forced Vibration of a Cable-Stayed Beam by Green’s Function Approach

Abstract

A method based on Green’s functions is proposed to study the steady-state dynamic responses of a cable-stayed beam subjected to distributed and/or concentrated loadings. The cable’s initial sag and damping effects are considered. Also, the quadratic and cubic nonlinearities due to the dynamic strain of the cable are taken into account. The nonlinear governing equations of the system are solved by the perturbation method, and the Laplace transformation is employed to derive the Green’s functions for the beam and cable with specified boundary conditions. Then the closed form solution of the linear system and the perturbation analytical solution of the nonlinear system are given in an integral form based on those Green’s functions. The difference between the Green’s functions for the linear and nonlinear systems is caused by the nonlinear boundary conditions. The natural frequencies of the cable-stayed beam calculated by the proposed method are compared with those in the literature to illustrate the validity of the present approach. Particularly, the cable’s dynamic strain effects on the symmetric property of the Green’s functions are discussed. The effects of the nonlinear terms on the amplitude of the Green’s functions and vibration are investigated, along with that of damping. The technique presented should find its applicability for other complex structures, such as cable-stayed bridges and steel structures.